AI 中文总结
研究将乘性卡尔德隆预处理器从低阶推广到高阶，利用基于B样条的基函数建立显式高阶对偶基，其GMRES迭代次数低且恒定，与未知量数量和多项式次数无关，为发挥高阶基潜力提供关键支持。
AI 中文摘要
用Rao-Wilton-Glisson（RWG）函数离散化电场积分方程（EFIE）时，最有效的预处理手段之一是采用Buffa-Christiansen（BC）函数作为RWG基对偶基的乘性卡尔德隆预处理器，可避免密集离散化和低频崩溃。为将乘性卡尔德隆预处理器从低阶BC和RWG基推广到高阶，利用基于B样条的基函数建立首个显式高阶对偶基，它是BC函数到任意多项式次数的推广。数值结果表明，所得预处理器的广义最小残差（GMRES）迭代次数低且恒定，与未知量数量和多项式次数无关，这是发挥高阶基全部潜力的关键。
英文摘要
One of the most effective means to precondition the electric field integral equation (EFIE) discretized with Rao-Wilton-Glisson (RWG) functions is the multiplicative Calderón preconditioner employing Buffa-Christiansen (BC) functions as a basis dual to the RWG basis. It results in a formulation that is free from the dense-discretization and the low-frequency breakdown. To generalize the multiplicative Calderón preconditioner from the low-order BC and RWG basis to higher orders, we utilize B-spline-based basis functions and establish the first explicit high-order dual basis. It can be regarded as a generalization of the BC functions to arbitrary polynomial degrees and constitutes a fundamental building block for other approaches that rely on a dual basis. Numerical results for the obtained preconditioner demonstrate a low and constant number of generalized minimum residual (GMRES) iterations independent of the number of unknonws and the polynomial degree for canonical and realistic perfectly electrically conducting (PEC) scatterers; a key to enable the full potential of higher-order bases.
